375 15.3, Signals with a One-sided Spectrum For time signals, the same definition of t,he Hilbert trmsforin as in (15.19) is true: Jg+ / 03 I ?-L{x(t)) = -n ?-L{x(t)) n1x ( t ) * -tt )( n t-z n -a! 3o (15.41) I Here the indepericient vtriablc is in contrast to (15.19) just called t and not w A shorter and significaiitly clearer way of carryirig out the Hilbcrt transform by calculating thc convolution integral (15.41) can be obtained by switching to the frequency-dornain Using the convolution property (9.70) on (15.41) yields (15.42) The transform pair (9.39) follows ~ { x ( t )=}F ( - j X ( j w ) sign ( w ) } (15.43) The Hilbert transform S ( t ) of a time signal ~ ( tis)obtained if a new spectrum is formed from the spectrum X ( j w ) by inverting the sign for w < and multiplying by - j The fiinction of time a($)corresponding to this spectrum is the Hilbert, transform of ~ ( t ) This procedure is already well known, if we think of the EIilbert t,ransforin of a time signal as the effect, of a system 7-l on the input signal z ( t ) (Fig 15.3) The fuiict,ion l/(nt) is then the impulse response of the system 3-1 and (15.41) describes the convolution of the input signal z(t) with the imp-iilse response The alternative calculation in the frequency-domain is the multiplicatioii of the input signal's spectrum X ( j w ) with the transfer function o fth e system 3-1 HE(jw) = -jsign(w) (15.44) This method using the freqiencg-domain is significantly simpler and safer to use t hari evaluating the convolution integral (15.41) Furthermore, exploiting duality this method can also be used on (15.19) and likewise on discreLe causal systems and uriilateral periodic spectra From (15.40), it can br seen that, a certain real part Re { h ( t ) ) ,also defines the corresponding imaginary part apart from an additive constant, if the spectrum has no contribution for U? < Therefore, from a real signal r(t) a new signal x1( t ) can he derived by (15.45) Xl(f) = (t) t 13-1{x(O} The relations (1 5.39) and( 15.40) hold for its real and imaginary components lt therefore has a one-sided spectrum X ~ ( j w )The , mean value of the imaginary part has been arbitrarily set to Ini{Ho} = 0; it coiild have been any other value